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On Frobenius Theorem and Classication of 2-Dimensional Real Division Algebras

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On Frobenius Theorem and Classication of 2-Dimensional Real Division Algebras U.U.D.M. Project Report 2020:18 On Frobenius Theorem and Classication of 2-Dimensional Real Division Algebras Mikolaj Cuszynski-Kruk Examensarbete i matematik, 15 hp Handledare: Martin Herschend Examinator: Veronica Crispin Quinonez Juni 2020 Department of Mathematics Uppsala University Abstract A proof of Frobenius theorem which states that the only finite-dimensional real associative division algebras up to isomorphism are R; C and H, is given. A com- plete list of all 2-dimensional real division algebras based on the multiplication table of the basis is given, based on the work of Althoen and Kugler. The list is irredundant in all cases except the algebras that have exactly 3 idempotent elements. Contents 1 Introduction 1 2 Background 2 2.1 Rings . 2 2.2 Linear algebra . 4 2.3 Algebra . 7 3 Frobenius Theorem 10 4 Classification of finite-dimensional real division algebras 12 4.1 2-dimensional division algebras . 13 References 22 1 Introduction An algebra A over a field F is vector space over the same field together with a bilinear multiplication. Moreover, an algebra is a division algebra if for all non-zero a in A the maps La : A ! A defined by v 7! av and Ra : A ! A defined by v 7! va are invertible. If an algebra has a unit and the multiplication is associative then the algebra has a ring structure. In this paper we will only study algebras over the field of real numbers which are called real algebras, moreover we shall assume that the vector space is finite-dimensional. A trivial example of a real division algebra is the field of real numbers with standard multiplication. The study of real division algebras originate form the study of number sys- tems. The construction of the field of complex numbers yielded a 2-dimensional real division algebra. Further developments were made by Sir William Rowan Hamilton. After studying the complex numbers he tried to construction a 3- dimensional real division algebra and according to a famous story, after many failed attempts, during a walk Hamilton came up with a 4-dimensional con- struction instead, defined by the property i2 = j2 = k2 = ijk = −1 which later would be called the quaternion algebra. Although Hamilton suc- ceeded in constructing a 4-dimensional division algebra there was a price to be paid, the quaternions are not commutative. For further history of Hamilton and the quaternions see [2]. In the same way complex numbers can be seen as pairs of real numbers with multiplication defined by (1), the quaternions can be see as pairs of complex numbers with multiplication also defined by (1). (z1; z2) · (w1; w2) = (z1w1 − w2z2; z2w1 + w2z1): (1) By continuing this line of thoughts, pairs of quaternions are a perfect candidate for a 8-dimensional real division algebra. Indeed, they form the so called octo- nions discovered independently by Graves [8] and Cayley [5]. Similarly to the quaternions not being commutative the octonions are in addition not associat- ive. One could think that it is possible to construct 2n−dimensional real division algebras by continuing this process, but it is not the case. In 1958 Bott and Milnor [4] proved that the only possible dimensions of a real division algebra are 1,2,4 and 8. In 1878 Frobenius proved that the only real associative division algebras up to isomorphism are R; C and H, a proof will be given in Section 3. Later in 1931, Zorn [12] proved that by weakening the assumption of associativity to instead only require that the algebra is alternative yields only one additional algebra, the octonion algebra. When considering arbitrary real division algebras, the classification of 2-di- mension real division algebras is known and will be given in Section 4.1. For 1 the 4- and 8-dimensional case only some special cases have been classified. An overview of some classifications is given in [6]. 2 Background In this section we define useful notions and establish results that will be neces- sary in order to prove Frobenius theorem. 2.1 Rings Basic theory of rings will be needed in order to understand properties of an associative algebra. The following subsection is based on [11, Chapter 2]. A natural way to start is to define the notion of a ring. Definition 2.1. A ring is a 5-tuple (R; +; ·; 0; 1) consisting of a set R, two binary operation + and · that are closed in R and two elements 0; 1 2 R such that for all a; b; c 2 R (a + b) + c = a + (b + c); (A1) a + b = b + a; (A2) a + 0 = 0 = 0 + a; (A3) 9 −a 2 R a + (−a) = 0 = −a + a; (A4) (a · b) · c = a · (b · c); (M1) a · 1 = a = 1 · a; (M2) a · (b + c) = a · b + a · c; (D1) (b + c) · a = b · a + c · a: (D2) Remark 2.1.1. (R; +; ·; 0; 1) will often be denoted simply by R and a · b by ab. Example 2.1.2. (C; +; ·; 0; 1) where the operations are interpreted as standard addition and multiplication of complex numbers, is a ring. Definition 2.2. (S; +; ·; 0; 1) is called a subring of a ring (R; +; ·; 0; 1) if S ⊆ R and the following holds for all a; b 2 S 0; 1 2 S; (S1) a + b 2 S; (S2) −a 2 S; (S3) ab 2 S: (S4) Remark 2.2.1. In order to see if S satisfies 0 2 S, (S2) and (S3) it is enough to check if S is non-empty and that for all x; y in S, x − y is also in S. Since then x in S implies that 0 = x − x is in S and for all x; y in S, −x = 0 − x is also in S and hence so is y + x = y − (−x). 2 Since Frobenius theorem concerns division algebras the ability to perform division is important. Division is defined as the inverse of multiplication but an ring does not required the existence of multiplicative inverses and rings that does have multiplicative inverses are called division rings. Definition 2.3. A division ring is a ring in which every non-zero element has an inverse, i.e. a ring R such that 8x 2 R n f0g 9y 2 R xy = 1 = yx: Proposition 2.4. For each x 2 R it's inverse is unique and denoted x−1. Proof. Assume y1; y2 are inverses of x, then y1x = 1 = xy2 and y1 = y1(xy2) = (y1x)y2 = y2. The quaternions were first constructed by Hamilton and can be seen as an extension of complex number by two additional imaginary units, j and k, that satisfy i2 = j2 = k2 = −1 = ijk. Hamilton's construction was of a geometrical nature but there is another way of constructing quaternions, namely as a subset of M2×2(C), i.e the set of 2 × 2 matrices with complex entries. z w Proposition 2.5. The set = : z; w 2 is a subring of M ( ). H −w z C 2×2 C Proof. For all z; w 2 C it holds that z − w = z − w hence if Z; W 2 H then 1 0 Z − W 2 . The multiplicative identity in M ( ) is I = and since H 2×2 C 0 1 x y z w 1 = 1 and 0 = 0, I is in . Moreover ; 2 implies H −y x −w z H x y z w xz − yw xw + yz xz − yw xw + yz = = −y x −w z −yz − xw −yw + xz −xw + yz xz − yw since z w = zw for all complex numbers. Hence H is closed under multiplication and thus a subring of M2×2(C). Remark 2.5.1. Note that every matrix in H is of the form p p x1 + x2 −1 x3 + x4 −1 H 3 p p −x3 + x4 −1 x1 − x2 −1 p p 1 0 −1 0 0 1 0 −1 = x + x p + x + x p 1 0 1 2 0 − −1 3 −1 0 4 −1 0 1 0 where x ; x ; x ; x 2 , moreover if we denote the matrices ; 1 2 3 4 R 0 1 p p −1 0 0 1 0 −1 p ; and p by 1; i; j and k, respectively, 0 − −1 −1 0 −1 0 then it can easily be checked that i2 = j2 = k2 = −1 = ijk holds. 3 Proposition 2.6. The ring of quaternions is a division ring. z w Proof. Let 2 n f0g then −w z H 1 z −w 1 z −w = 2 H and jzj2 + jwj2 w z jzj2 + jwj2 −−w z 1 z w z −w 1 0 = : jzj2 + jwj2 −w z w z 0 1 Remark 2.6.1. Note that the equation i2 = j2 = k2 = −1 = ijk determine the multiplication in H completely. Definition 2.7. The centre of a ring R is the subset Z(R) = fa 2 R : 8b 2 R; ab = bag: Lemma 2.8. If r 2 R then r1 2 Z(H). r 0 Proof. Let r 2 then r1 = and for all z; w 2 , R 0 r C r 0 z w rz rw z w r 0 = = : 0 r −w z −rw rz −w z 0 r The notion of a fields will be useful in the next subsection and the following lemma is a technical one. Definition 2.9. A field is a non-zero division ring in which the multiplication commutes, i.e. a division ring R 6= f0g such that 8x; y 2 R; xy = yx.
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